I believe the chlorine will be quite bad for it, similar to leaving it in the sun for a long time or pouring bleach (corrosive) on it, as that's what household bleach actually is - a strong chlorine based solution.

Depends on how long ago the pool was chlorinated. Normal tap water is also chlorinated, and we all wash our gear with it. I'm guessing that as long as the pool doesn't smell of chlorine, or has a soapy feel to it, you'll be fine. the pH of a pool is usually not that far from neutral, say only 1/2 a point off.

I seem to remember they did a test on ropes - dippping them in water, coke - all sorts of things and then testing the strength. Interestingly enough, the rope dipped in urine ended up stronger. So now you know what to do...